Subgroups — #27277755

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272549	C₂	1	4	yes	maximal
4	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	2	yes	—
6	C₂×C₄×#88322	C₄	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₄×#88322	C₄	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	2	yes	—
10	#16810378	—	2	2	no	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#336	—	2	2	no	—
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂²×#88322	C₂×C₄	1	2	yes	—
17	C₄⋊#88322	—	2	4	no	—
18	C₄×#88322	—	2	2	no	—
19	C₄×#88322	—	2	2	no	—
20	C₄×#88322	—	2	2	no	—
21	C₄×#88322	—	2	4	no	—
22	C₄×#88322	C₈	1	4	yes	complemented
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	4	no	—
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	—	4	4	no	—
37	C₄²×C₅₃	—	2	4	no	—
38	C₄²×C₅₃	—	2	2	no	—
39	C₄²×C₅₃	—	2	4	no	—
40	C₄²×C₅₃	C₂×C₈	1	2	yes	—
41	C₄²×C₅₃	M₁₆	1	2	yes	—
42	C₄²×C₅₃	#41	1	2	yes	—
43	C₄²×C₅₃	#41	1	2	yes	—
44	C₂×#88322	—	2	2	no	—
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	—	2	2	no	—
47	C₂×#88322	—	2	2	no	—
48	C₂×#88322	—	4	2	no	—
49	C₂×#88322	—	4	4	no	—
50	C₂×#88322	C₂×C₈	1	4	yes	—
51	C₂×#88322	M₁₆	1	4	yes	—
52	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₅₃	#108	1	4	yes	—
60	C₂×C₄×C₅₃	#117	1	2	yes	—
61	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#125	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#135	1	4	yes	—
68	C₈×C₅₃	—	8	2	no	—
69	#88322	—	2	4	no	—
70	#88322	—	2	4	no	—
71	#88322	—	8	2	no	—
72	#8785	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	#382	1	1	yes	characteristic
74	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₂²×C₅₃	#423	1	1	yes	characteristic
76	C₂²×C₅₃	#427	1	2	yes	—
77	C₂²×C₅₃	C₂×#108	1	2	yes	—
78	C₄×C₅₃	—	2	4	no	—
79	C₄×C₅₃	—	2	4	no	—
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	#431	1	2	yes	—
82	C₄×C₅₃	#432	1	2	yes	—
83	C₄×C₅₃	#488	1	2	yes	—
84	C₄×C₅₃	#488	1	2	yes	—
85	#1831	—	53	1	no	—
86	#1831	D₅₃	1	1	yes	characteristic
87	C₄²×C₈	—	53	1	no	—
88	#1277	—	53	4	no	—
89	C₂×C₅₃	#1564	1	2	yes	—
90	C₂×C₅₃	#1568	1	2	yes	—
91	C₂×C₅₃	#1831	1	1	yes	characteristic
92	C₂×C₅₃	#1835	1	1	yes	characteristic
93	C₂×C₅₃	#1837	1	1	yes	characteristic
94	C₂×C₄⋊C₈	—	53	2	no	—
95	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
96	C₂×C₄×C₈	—	53	1	no	—
97	C₂×C₄×C₈	—	53	2	no	—
98	C₂×C₄×C₈	—	53	1	no	—
99	C₂×C₄×C₈	—	53	2	no	—
100	C₄³	C₂×D₅₃	1	1	yes	characteristic
101	#336	—	2	2	no	—
102	#336	—	106	2	no	—
103	C₅₃	#8785	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	53	2	no	—
106	C₂²×C₈	—	53	2	no	—
107	C₂²×C₈	—	106	2	no	—
108	C₂×C₄²	Dic₁₀₆	1	2	yes	—
109	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
110	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
111	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
112	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	106	4	no	—
115	C₄×C₈	—	53	4	no	—
116	C₄×C₈	—	106	4	no	—
117	C₄×C₈	—	106	2	no	—
118	C₄×C₈	—	106	2	no	—
119	C₄×C₈	—	106	2	no	—
120	C₂²×C₄	#11719919	1	2	yes	—
121	C₂²×C₄	#11719919	1	2	yes	—
122	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	4	4	no	—
127	C₂×C₈	—	53	4	no	—
128	C₂×C₈	—	53	4	no	—
129	C₂×C₈	—	106	2	no	—
130	C₂×C₈	—	106	2	no	—
131	C₂×C₈	—	106	2	no	—
132	C₂×C₈	—	106	2	no	—
133	C₂×C₈	—	212	2	no	—
134	C₂×C₈	—	212	4	no	—
135	C₄²	—	2	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	4	no	—
138	C₄²	C₈×D₅₃	1	2	yes	—
139	C₄²	#11719912	1	2	yes	—
140	C₄²	#11719947	1	2	yes	—
141	C₄²	#11719949	1	2	yes	—
142	C₂³	C₂×#11719919	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	#13133146	1	4	yes	—
149	C₂×C₄	#13133152	1	4	yes	—
150	C₂×C₄	#13133175	1	2	yes	—
151	C₂×C₄	#13133204	1	1	yes	characteristic
152	C₂×C₄	#13133206	1	1	yes	characteristic
153	C₂×C₄	#13133208	1	2	yes	—
154	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
155	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
156	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
157	C₂×C₄	#13133237	1	4	yes	—
158	C₈	—	8	2	no	—
159	C₈	—	106	4	no	—
160	C₈	—	106	4	no	—
161	C₈	—	424	2	no	—
162	C₂²	#16810488	1	1	yes	characteristic
163	C₂²	C₂×#13133146	1	1	yes	characteristic
164	C₂²	#16810646	1	1	yes	characteristic
165	C₂²	#16810659	1	2	yes	—
166	C₂²	C₂×#13133152	1	2	yes	—
167	C₄	—	2	4	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	#16810675	1	2	yes	—
171	C₄	#16810679	1	2	yes	—
172	C₄	#16810977	1	2	yes	—
173	C₄	#16810980	1	2	yes	—
174	C₂	#20273461	1	2	yes	—
175	C₂	#20273476	1	2	yes	—
176	C₂	#20274831	1	1	yes	characteristic
177	C₂	#20274845	1	1	yes	characteristic
178	C₂	#20274852	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#8785
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
22. C₄×#88322	C₈	semidirect
103. C₅₃	#8785	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20274832	2
C₄²×#88322	2
#20272549	2
C₅₃×#1831	2
#8785	53